FRP.Elerea.Simple

Contents

Description

This module provides leak-free and referentially transparent higher-order discrete signals.

Synopsis

The signal abstraction

A signal represents a value changing over time. It can be thought of as a function of type Nat -> a, where the argument is the sampling time, and the Monad instance agrees with the intuition (bind corresponds to extracting the current sample). Signals and the values they carry are denoted the following way in the documentation:

 s = <<s0 s1 s2 ...>>

This says that s is a signal that reads s0 during the first sampling, s1 during the second and so on. You can also think of s as the following function:

 s t_sample = [s0,s1,s2,...] !! t_sample

Signals are constrained to be sampled sequentially, there is no random access. The only way to observe their output is through start.

Instances

Monad Signal
Functor Signal
Applicative Signal
Bounded t => Bounded (Signal t)
Enum t => Enum (Signal t)
Eq (Signal a)	Equality test is impossible.
Floating t => Floating (Signal t)
Fractional t => Fractional (Signal t)
Integral t => Integral (Signal t)
Num t => Num (Signal t)
Ord t => Ord (Signal t)
Real t => Real (Signal t)
Show (Signal a)	The Show instance is only defined for the sake of Num...

data SignalGen a Source

A signal generator is the only source of stateful signals. It can be thought of as a function of type Nat -> a, where the result is an arbitrary data structure that can potentially contain new signals, and the argument is the creation time of these new signals. It exposes the MonadFix interface, which makes it possible to define signals in terms of each other. The denotation of signal generators happens to be the same as that of signals, but this partly accidental (it does not hold in the other variants), so we will use a separate notation for generators:

 g = <|g0 g1 g2 ...|>

Just like signals, generators behave as functions of time:

 g t_start = [g0,g1,g2,...] !! t_start

The conceptual difference between the two notions is that signals are passed a sampling time, while generators expect a start time that will be the creation time of all the freshly generated signals in the resulting structure.

Instances

Monad SignalGen
Functor SignalGen
MonadFix SignalGen
Applicative SignalGen

Embedding into I/O

start Source

Arguments

:: SignalGen (Signal a)	the generator of the top-level signal
-> IO (IO a)	the computation to sample the signal

Embedding a signal into an IO environment. Repeated calls to the computation returned cause the whole network to be updated, and the current sample of the top-level signal is produced as a result. This is the only way to extract a signal generator outside the network, and it is equivalent to passing zero to the function representing the generator. In general:

 replicateM n =<< start <|<<x0 x1 x2 x3 ...>> ...|> == take n [x0,x1,x2,x3,...]

Example:

 do
     smp <- start (stateful 3 (+2))
     res <- replicateM 5 smp
     print res

Output:

 [3,5,7,9,11]

external Source

Arguments

:: a	initial value
-> IO (Signal a, a -> IO ())	the signal and an IO function to feed it

A signal that can be directly fed through the sink function returned. This can be used to attach the network to the outer world. The signal always yields the value last written to the sink. In other words, if the sink is written less frequently than the network sampled, the output remains the same during several samples. If values are pushed in the sink more frequently, only the last one before sampling is visible on the output.

Example:

 do
     (sig,snk) <- external 4
     smp <- start (return sig)
     r1 <- smp
     r2 <- smp
     snk 7
     r3 <- smp
     snk 9
     snk 2
     r4 <- smp
     print [r1,r2,r3,r4]

Output:

 [4,4,7,2]

externalMulti Source

Arguments

:: IO (SignalGen (Signal [a]), a -> IO ())

a generator for the event signal and the associated sink

An event-like signal that can be fed through the sink function returned. The signal carries a list of values fed in since the last sampling, i.e. it is constantly [] if the sink is never invoked. The order of elements is reversed, so the last value passed to the sink is the head of the list. Note that unlike external this function only returns a generator to be used within the expression constructing the top-level stream, and this generator can only be used once.

Example:

 do
     (gen,snk) <- externalMulti
     smp <- start gen
     r1 <- smp
     snk 7
     r2 <- smp
     r3 <- smp
     snk 9
     snk 2
     r4 <- smp
     print [r1,r2,r3,r4]

Output:

 [[],[7],[],[2,9]]

Basic building blocks

delay Source

Arguments

:: a	initial output at creation time
-> Signal a	the signal to delay
-> SignalGen (Signal a)	the delayed signal

The delay combinator is the elementary building block for adding state to the signal network by constructing delayed versions of a signal that emit a given value at creation time and the previous output of the signal afterwards (-- is undefined):

 delay x0 s = <| <<x0 s0 s1 s2 s3 ...>>
                 <<-- x0 s1 s2 s3 ...>>
                 <<-- -- x0 s2 s3 ...>>
                 <<-- -- -- x0 s3 ...>>
                 ...
              |>

It can be thought of as the following function (which should also make it clear why the return value is SignalGen):

 delay x0 s t_start t_sample
   | t_start == t_sample = x0
   | t_start < t_sample  = s (t_sample-1)
   | otherwise           = error \"Premature sample!\"

The way signal generators are extracted by generator ensures that the error can never happen.

Example (requires the DoRec extension):

 do
     smp <- start $ do
         rec let fib'' = liftA2 (+) fib' fib
             fib' <- delay 1 fib''
             fib <- delay 1 fib'
         return fib
     res <- replicateM 7 smp
     print res

Output:

 [1,1,2,3,5,8,13]

snapshot :: Signal a -> SignalGen aSource

A formal conversion from signals to signal generators, which effectively allows for retrieving the current value of a previously created signal within a generator. This includes both signals defined in an external scope as well as those created earlier in the same generator. In the model, it corresponds to the identity function.

generator Source

Arguments

:: Signal (SignalGen a)	the signal of generators to run
-> SignalGen (Signal a)	the signal of generated structures

A reactive signal that takes the value to output from a signal generator carried by its input with the sampling time provided as the start time for the generated structure. It is possible to create new signals in the monad, which is the key to defining dynamic data-flow networks.

 generator << <|x00 x01 x02 ...|>
              <|x10 x11 x12 ...|>
              <|x20 x21 x22 ...|>
              ...
           >> = <| <<x00 x11 x22 ...>>
                   <<x00 x11 x22 ...>>
                   <<x00 x11 x22 ...>>
                   ...
                |>

It can be thought of as the following function:

 generator g t_start t_sample = g t_sample t_sample

It has to live in the SignalGen monad, because it needs to maintain an internal state to be able to cache the current sample for efficiency reasons. However, this state is not carried between samples, therefore start time doesn't matter and can be ignored.

Refer to the longer example at the bottom to see how it can be used.

memo Source

Arguments

:: Signal a	the signal to cache
-> SignalGen (Signal a)	a signal observationally equivalent to the argument

Memoising combinator. It can be used to cache results of applicative combinators in case they are used in several places. It is observationally equivalent to return in the SignalGen monad.

 memo s = <|s s s s ...|>

For instance, if s = f <$> s', then f will be recalculated once for each sampling of s. This can be avoided by writing s <- memo (f <$> s') instead. However, memo incurs a small overhead, therefore it should not be used blindly.

All the functions defined in this module return memoised signals.

until Source

Arguments

:: Signal Bool	the boolean input signal
-> SignalGen (Signal Bool)	a one-shot signal true only the first time the input is true

A signal that is true exactly once: the first time the input signal is true. Afterwards, it is constantly false, and it holds no reference to the input signal. For instance (assuming the rest of the input is constantly False):

 until <<False False True True False True ...>> =
     <| <<False False True  False False False False False False False ...>>
        << ---  False True  False False False False False False False ...>>
        << ---   ---  True  False False False False False False False ...>>
        << ---   ---   ---  True  False False False False False False ...>>
        << ---   ---   ---   ---  False True  False False False False ...>>
        << ---   ---   ---   ---   ---  True  False False False False ...>>
        << ---   ---   ---   ---   ---   ---  False False False False ...>>
        ...
     |>

It is observationally equivalent to the following expression (which would hold onto s forever):

 until s = do
     step <- transfer False (||) s
     dstep <- delay False step
     memo (liftA2 (/=) step dstep)

Example:

 do
     smp <- start $ do
         cnt <- stateful 0 (+1)
         tick <- until ((>=3) <$> cnt)
         return $ liftA2 (,) cnt tick
     res <- replicateM 6 smp
     print res

Output:

 [(0,False),(1,False),(2,False),(3,True),(4,False),(5,False)]

Derived combinators

stateful Source

Arguments

:: a	initial state
-> (a -> a)	state transformation
-> SignalGen (Signal a)

A pure stateful signal. The initial state is the first output, and every subsequent state is derived from the preceding one by applying a pure transformation.

Example:

 do
     smp <- start (stateful "x" ('x':))
     res <- replicateM 5 smp
     print res

Output:

 ["x","xx","xxx","xxxx","xxxxx"]

transfer Source

Arguments

:: a	initial internal state
-> (t -> a -> a)	state updater function
-> Signal t	input signal
-> SignalGen (Signal a)

A stateful transfer function. The current input affects the current output, i.e. the initial state given in the first argument is considered to appear before the first output, and can never be observed, and subsequent states are determined by combining the preceding state with the current output of the input signal using the function supplied.

Example:

 do
     smp <- start $ do
         cnt <- stateful 1 (+1)
         transfer 10 (+) cnt
     res <- replicateM 5 smp
     print res

Output:

 [11,13,16,20,25]

transfer2 Source

Arguments

:: a	initial internal state
-> (t1 -> t2 -> a -> a)	state updater function
-> Signal t1	input signal 1
-> Signal t2	input signal 2
-> SignalGen (Signal a)

A variation of transfer with two input signals.

transfer3 Source

Arguments

:: a	initial internal state
-> (t1 -> t2 -> t3 -> a -> a)	state updater function
-> Signal t1	input signal 1
-> Signal t2	input signal 2
-> Signal t3	input signal 3
-> SignalGen (Signal a)

A variation of transfer with three input signals.

transfer4 Source

Arguments

:: a	initial internal state
-> (t1 -> t2 -> t3 -> t4 -> a -> a)	state updater function
-> Signal t1	input signal 1
-> Signal t2	input signal 2
-> Signal t3	input signal 3
-> Signal t4	input signal 4
-> SignalGen (Signal a)

A variation of transfer with four input signals.

Signals with side effects

The following combinators are primarily aimed at library implementors who wish build abstractions to effectful libraries on top of Elerea.

execute :: IO a -> SignalGen aSource

An IO action executed in the SignalGen monad. Can be used as liftIO.

effectful Source

Arguments

:: IO a	the action to be executed repeatedly
-> SignalGen (Signal a)

A signal that executes a given IO action once at every sampling.

In essence, this combinator provides cooperative multitasking capabilities, and its primary purpose is to assist library writers in wrapping effectful APIs as conceptually pure signals. If there are several effectful signals in the system, their order of execution is undefined and should not be relied on.

Example:

 do
     smp <- start $ do
         ref <- execute $ newIORef 0
         effectful $ do
             x <- readIORef ref
             putStrLn $ "Count: " ++ show x
             writeIORef ref $! x+1
             return ()
     replicateM_ 5 smp

Output:

 Count: 0
 Count: 1
 Count: 2
 Count: 3
 Count: 4

Another example (requires mersenne-random):

 do
     smp <- start $ effectful randomIO :: IO (IO Double)
     res <- replicateM 5 smp
     print res

Output:

 [0.12067753390401374,0.8658877349182655,0.7159264443196786,0.1756941896012891,0.9513646060896676]

effectful1 Source

Arguments

:: (t -> IO a)	the action to be executed repeatedly
-> Signal t	parameter signal
-> SignalGen (Signal a)

A signal that executes a parametric IO action once at every sampling. The parameter is supplied by another signal at every sampling step.

effectful2 Source

Arguments

:: (t1 -> t2 -> IO a)	the action to be executed repeatedly
-> Signal t1	parameter signal 1
-> Signal t2	parameter signal 2
-> SignalGen (Signal a)

Like effectful1, but with two parameter signals.

effectful3 Source

Arguments

:: (t1 -> t2 -> t3 -> IO a)	the action to be executed repeatedly
-> Signal t1	parameter signal 1
-> Signal t2	parameter signal 2
-> Signal t3	parameter signal 3
-> SignalGen (Signal a)

Like effectful1, but with three parameter signals.

effectful4 Source

Arguments

:: (t1 -> t2 -> t3 -> t4 -> IO a)	the action to be executed repeatedly
-> Signal t1	parameter signal 1
-> Signal t2	parameter signal 2
-> Signal t3	parameter signal 3
-> Signal t4	parameter signal 4
-> SignalGen (Signal a)

Like effectful1, but with four parameter signals.

A longer example

For a not entirely trivial example, let's create a dynamic collection of countdown timers, where each expired timer is removed from the collection. First of all, we'll need a simple tester function:

 sigtest gen = replicateM 15 =<< start gen

We can try it with a trivial example:

 > sigtest $ stateful 2 (+3)
 [2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47]

Our first definition will be a signal representing a simple named timer:

 countdown :: String -> Int -> SignalGen (Signal (String,Maybe Int))
 countdown name t = do
   let tick prev = do { t <- prev ; guard (t > 0) ; return (t-1) }
   timer <- stateful (Just t) tick
   return ((,) name <$> timer)

Let's see if it works:

 > sigtest $ countdown "foo" 4
 [("foo",Just 4),("foo",Just 3),("foo",Just 2),("foo",Just 1),("foo",Just 0),
  ("foo",Nothing),("foo",Nothing),("foo",Nothing),...]

Next, we will define a timer source that takes a list of timer names, starting values and start times and creates a signal that delivers the list of new timers at every point:

 timerSource :: [(String, Int, Int)] -> SignalGen (Signal [Signal (String, Maybe Int)])
 timerSource ts = do
   let gen t = mapM (uncurry countdown) newTimers
           where newTimers = [(n,v) | (n,v,st) <- ts, st == t]
   cnt <- stateful 0 (+1)
   generator (gen <$> cnt)

Now we need to encapsulate the timer source signal in another signal expression that takes care of maintaining the list of live timers. Since working with dynamic collections is a recurring task, let's define a generic combinator that maintains a dynamic list of signals given a source and a test that tells from the output of each signal whether it should be kept. We can use mdo expressions (a variant of do expressions allowing forward references) as syntactic sugar for mfix to make life easier:

 collection :: Signal [Signal a] -> (a -> Bool) -> SignalGen (Signal [a])
 collection source isAlive = mdo
   sig <- delay [] (map snd <$> collWithVals')
   coll <- memo (liftA2 (++) source sig)
   let collWithVals = zip <$> (sequence =<< coll) <*> coll
   collWithVals' <- memo (filter (isAlive . fst) <$> collWithVals)
   return $ map fst <$> collWithVals'

We need recursion to define the coll signal as a delayed version of its continuation, which does not contain signals that need to be removed in the current sample. At every point of time the running collection is concatenated with the source. We define collWithVals, which simply pairs up every signal with its current output. The output is obtained by extracting the current value of the signal container and sampling each element with sequence. We can then derive collWithVals', which contains only the signals that must be kept for the next round along with their output. Both coll and collWithVals' have to be memoised, because they are used more than once (the program would work without that, but it would recalculate both signals each time they are used). By throwing out the respective parts, we can get both the final output and the collection for the next step (coll').

Now we can easily finish the original task:

 timers :: [(String, Int, Int)] -> SignalGen (Signal [(String, Int)])
 timers timerData = do
   src <- timerSource timerData
   getOutput <$> collection src (isJust . snd)
     where getOutput = fmap (map (\(name,Just val) -> (name,val)))

As a test, we can start four timers: a at t=0 with value 3, b and c at t=1 with values 5 and 3, and d at t=3 with value 4:

 > sigtest $ timers [("a",3,0),("b",5,1),("c",3,1),("d",4,3)]
 [[("a",3)],[("b",5),("c",3),("a",2)],[("b",4),("c",2),("a",1)],
  [("d",4),("b",3),("c",1),("a",0)],[("d",3),("b",2),("c",0)],
  [("d",2),("b",1)],[("d",1),("b",0)],[("d",0)],[],[],[],[],[],[],[]]

If the noise of the applicative lifting operators feels annoying, she (http://personal.cis.strath.ac.uk/~conor/pub/she/) comes to the save. Among other features it provides idiom brackets, which can substitute the explicit lifting. For instance, it allows us to define collection this way:

 collection :: Stream [Stream a] -> (a -> Bool) -> StreamGen (Stream [a])
 collection source isAlive = mdo
   sig <- delay [] (|map ~snd collWithVals'|)
   coll <- memo (|source ++ sig|)
   collWithVals' <- memo (|filter ~(isAlive . fst) (|zip (sequence =<< coll) coll|)|)
   return (|map ~fst collWithVals'|)